Package 'fsdaR'

An S3 object of class tclusticsol.object, (output of tclustICsol) containing the relevant solutions.

Wheather to include or not spurious solutions. By default spurios solutions are not included into the plot.

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

An object of S3 class returned by one of the estimation functions with the monitoring option selected (monitoring=TRUE): fsreda.object, sregeda.object, mmregeda.object, fsmeda.object, smulteda.object or mmmulteda.object. This is a list containing the monitoring of minimum Mahalanobis distance in case of multivariate analysis or the monitoring of residuals in case of regression.

The needed elements of out are

1. MAL: matrix containing the squared Mahalanobis distances monitored in each step of the forward search. Every row is associated with a unit (row of data matrix Y). This matrix can be created using one of the functions fsmult, smult or mmmult with the monitoring option selected, i.e. monitoring=TRUE.

2. RES: matrix containing the residuals monitored in each step of the forward search. Every row is associated with a unit (row of data matrix Y). This matrix can be created using the function fsreg with the monitoring option selected, i.e. monitoring=TRUE .

3. bdp: a vector containing breakdown point values that have been used, in case of S estimates.

4. eff: a vector containing efficiency values that have been used, in case of MM estimates.

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

An object of S3 class fsmeda.object returned by fsmult with monitoring=TRUE - a list containing the monitoring of minimum Mahalanobis distance.

The needed elements of out are

1. S2cov: matrix containing the monitoring of the elements of the covariance matrix in each step of the forward search:

2. Un: matrix containing the order of entry of each unit (necessary if datatooltip or databrush is selected).

3. X: The data matrix.

Controls the x scale in the plot. xlim is a vector with two elements controlling minimum and maximum on the x-axis. Default is to use automatic scale.

Controls the y scale in the plot. ylim is a vector with two elements controlling minimum and maximum on the y-axis. Default is to use automatic scale.

A title for the x axis

A title for the y axis

An overall title for the plot

The line width, a positive number, defaulting to 1

The line type. Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

Colors to be used for the highlighted units

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Numeric vector containing the subset size with length equal to the number of columns of matrix of mahalanobis distances. The default value of subsize is (nrow(MAL) - ncol(MAL) + 1):nrow(MAL)

(alternative to fg.unit) numeric vector of length 1 or 2 which specifies the highlighted trajectories. If length(fg.thresh) == 1 the highlighted trajectories are those of units that throughtout the search had at leat once a mahalanobis distance greater than fg.thresh. The default value is fg.thresh=2.5. If length(fg.thresh) == 2 the highlighted trajectories are those of units that throughtout the search had a mahalanobis distance at least once bigger than fg.thresh[2] or smaller than fg.thresh[1].

(alternative to fg.thresh), vector containing the list of the units to be highlighted. If fg.unit is supplied, fg.thresh is ignored.

numeric vector which specifies the steps of the search where to put labels for the highlighted trajectories (units). The default is to put the labels at the initial and final steps of the search. fg.labstep='' means no label.

The line width for the highlighted trajectories (units). Default is 1.

The line type for the highlighted trajectories (units). Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

colors to be used for the highlighted units.

Controlls whether to plot highlighted trajectories as symbols. if fg.mark==TRUE each line is plotted using a different symbol else no marker is used (default).

Controls the font size of the labels of the trajectories in foreground. If fg.cex=0 no labels will be shown - equivalent to fg.labstop="".

MATLAB-style arguments - appearance of the plot in terms of xlim, ylim, axes labels and their font size style, color of the lines, etc.

MATLAB-style arguments - for the trajectories in foregroud.

Plot handle. String which identifies the handle of the plot which is about to be created. The default is tag='pl_mmd'. Notice that if the program finds a plot which has a tag equal to the one specified by the user, then the output of the new plot overwrites the existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to have information about the unit selected, the step in which the unit enters the search and the associated label. If datatooltip is a list, it is possible to control the aspect of the data cursor (see MATLAB function datacursormode() for more details or see the examples below). The default options are DisplayStyle="Window" and SnapToDataVertex="on".

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

If rew=TRUE all subsequent output refers to reweighted else no reweighting is done.

p-by-1 vector containing the estimated regression parameters.

p x 1 vector containing the units forming subset associated with bLMS (bLTS).

residuals.

scale estimate of the residuals.

Vector like y containing weights. The elements of this vector are 0 or 1. These weights identify the h observations which are used to compute the final LTS (LMS) estimate. sum(weights)=h if there is not a perfect fit otherwise sum(weights) can be greater than h

The number of observations that have determined the LTS (LMS) estimator, i.e. the value of h.

vector containing the list of the units declared as outliers using confidence level specified in input scalar conflev.

confidence level which is used to declare outliers. Remark: conflev will be used to draw the horizontal lines (confidence bands) in the plots. Default value is 0.975

Number of subsets wihtout full rank. Notice that if this number is greater than 0.1*(number of subsamples) a warning is produced

the data matrix X

the response vector y

If rew=TRUE all subsequent output refers to reweighted else no reweighting is done.

p-by-1 vector containing the estimated regression parameters.

p x 1 vector containing the units forming subset associated with bLMS (bLTS).

residuals.

scale estimate of the residuals.

Vector like y containing weights. The elements of this vector are 0 or 1. These weights identify the h observations which are used to compute the final LTS (LMS) estimate. sum(weights)=h if there is not a perfect fit otherwise sum(weights) can be greater than h

The number of observations that have determined the LTS (LMS) estimator, i.e. the value of h.

vector containing the list of the units declared as outliers using confidence level specified in input scalar conflev.

confidence level which is used to declare outliers. Remark: conflev will be used to draw the horizontal lines (confidence bands) in the plots. Default value is 0.975

Number of subsets wihtout full rank. Notice that if this number is greater than 0.1*(number of subsamples) a warning is produced

the data matrix X

the response vector y

An n x p data matrix (n observations and p variables). Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

Plots the random starts minimum Mahalanobis distance with 1 If plot=FALSE (default) or plot=0 no plot is produced. The scale (ylim) for the y axis is defined as follows:

• ylim[2] is the maximum between the values of mmd in steps [n*0.2 n] and the final value of the 99 per cent envelope multiplied by 1.1.

• ylim[1] is the minimum between the values of mmd in steps [n*0.2 n] and the 1 per cent envelope multiplied by 0.9.

Remark: the plot which is produced is very simple. In order to control a series of options in this plot (including the y scale) and in order to connect it dynamically to the other forward plots it is necessary to use function mmdrsplot.

Point where to start monitoring required diagnostics. If init is not specified it will be set equal to (p+1).

A vector which specifies for which steps of the forward search it is necessary to save the units forming subset for each random start. if bsbsteps = 0 for each random start we store the units forming subset in all steps. The default is store the units forming subset in all steps if n <= 500 else to store the units forming subset at step init and steps which are multiple of 100. For example, if n = 753 and init = 6, units forming subset are stored for m=init, 100, 200, 300, 400, 500 and 600.

REMARK: The vector bsbsteps must contain numbers from init to n. if min(bsbsteps) < init a warning message will be issued.

Number of random starts. Default value is nsimul=200.

It controls whether to perform checks on matrix Y. If nocheck=TRUE, no check is performed.

If numpool > 1, the routine automatically checks if the Parallel Computing Toolbox is installed and distributes the random starts over numpool parallel processes. If numpool <= 1, the random starts are run sequentially. By default, numpool is set equal to the number of physical cores available in the CPU (this choice may be inconvenient if other applications are running concurrently). The same happens if the numpool value chosen by the user exceeds the available number of cores.

REMARK: up to R2013b, there was a limitation on the maximum number of cores that could be addressed by the parallel processing toolbox (8 and, more recently, 12). From R2014a, it is possible to run a local cluster of more than 12 workers.

REMARK: Unless you adjust the cluster profile, the default maximum number of workers is the same as the number of computational (physical) cores on the machine.

REMARK: In modern computers the number of logical cores is larger than the number of physical cores. By default, MATLAB is not using all logical cores because, normally, hyper-threading is enabled and some cores are reserved to this feature.

REMARK: It is because of Remarks 3 that we have chosen as default value for numpool the number of physical cores rather than the number of logical ones. The user can increase the number of parallel pool workers allocated to the multiple start monitoring by:

• setting the NumWorkers option in the local cluster profile settings to the number of logical cores (Remark 2). To do so go on the menu Home|Parallel|Manage Cluster Profile and set the desired "Number of workers to start on your local machine".

• setting numpool to the desired number of workers

Therefore, *if a parallel pool is not already open*, UserOption numpool (if set) overwrites the number of workers set in the local/current profile. Similarly, the number of workers in the local/current profile overwrites default value of numpool obtained as feature('numCores') (i.e. the number of physical cores).

Set cleanpool cleanpool=TRUE if the parallel pool has to be cleaned after the execution of the random starts. Otherwise (default) cleanpool=FALSE. Clearly this option has an effect just if previous option numpool > 1.

Level of output to sidplay. It controls whether to display or not messages about random start progress. More precisely, if previous option numpool > 1, then a progress bar is displayed, on the other hand a message will be displayed on the screen when 10

REMARK: in order to create the progress bar when nparpool > 1 the program writes on a temporary .txt file in the folder where the user is working. Therefore it is necessary to work in a folder where the user has write permission. If this is not the case and the user (say) is working without write permission in folder C:/Program Files/MATLAB the following message will appear on the screen:

Error using ProgressBar (line 57) Do you have write permissions for C:/Program Files/MATLAB?"

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

An n x p data matrix (n observations and p variables). Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

List of units forming the initial subset or size of the initial subset. If monitoring=FALSE the default is to start the search with p+1 units, containing those observations which are not outlying on any scatterplot, found as the intersection of all points lying within a robust contour containing a specified portion of the data (Riani and Zani 1997) and inside the univariate boxplot.

Remark: if bsb is a vector, the option crit is ignored.

Wheather to perform monitoring of Mahalanobis distances and other specific quantities

If specified, the criterion to be used to initialize the search.

• If crit="md" the units which form initial subset are those which have the smallest m0 pseudo Mahalanobis distances computed using procedure unibiv() (bivariate robust ellipses).

• If crit="biv" sorting is done first in terms of times units fell outside robust bivariate ellipses and then in terms of pseudoMD. In other words, the units forming initial subset are chosen first among the set of those which never fell outside robust bivariate ellipses then among those which fell only once outside bivariate ellipses ... up to reach m0.

• If crit="uni" sorting is done first in terms of times units fell outside univariate boxplots and then in terms of pseudoMD. In other words, the units forming initial subset are chosen first among the set of those which never fell outside univariate boxplots then among those which fell only once outside univariate boxplots... up to reach m0.

Remark: as the user can see the starting point of the search is not going to affect at all the results of the analysis. The user can explore this point with his own datasets.

Remark: if crit="biv" the user can also supply in scalar rf (see below) the confidence level of the bivariate ellipses.

Confidence level for bivariate ellipses. The default is 0.95. This option is useful only if crit='biv'.

Point where to start monitoring required diagnostics. Note that if a vector m0 is supplied, init >= length(m0). If init is not specified it will be set equal to floor(n*0.6).

Plots the minimum Mahalanobis distance. If plot=FALSE (default) or plot=0 no plot is produced. If plot=TRUE the plot of minimum MD with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. If plot=2 the additional plots of envelope resuperimposition are produced. If plot is a list it may contain the following fields:

• ylim vector with two elements controlling minimum and maximum on the y axis. Default value is ” (automatic scale)

• xlim vector with two elements controlling minimum and maximum on the x axis. Default value is ” (automatic scale)

• resuper vector which specifies for which steps it is necessary to show the plots of resuperimposed envelopes if resuper is not supplied a plot of each step in which the envelope is resuperimposed is shown. Example: if resuper = c(85 87) plots of resuperimposedenvelopes are shown at steps m=85 and m=87

• ncoord If ncoord=1 plots are shown in normal coordinates else (default) plots are shown in traditional mmd coordinates

• labeladd If labeladd=1, the outliers in the spm are labelled with the unit row index. The default value is labeladd="", i.e. no label is added

• nameY character vector containing the labels of the variables. As default value, the labels which are added are Y1, ...Yp.

• lwd controls line width of the curve which contains the monitoring of minimum Mahalanobis distance. Default is lwd=2.

• lwdenv Controls linewidth of the envelopes. Default is lwdenv=2.

Option that might be used to identify extreme outliers when the distribution of the data is strongly non normal. In these circumstances, the general signal detection rule based on consecutive exceedances cannot be used. In this case bonflev can be:

1. a scalar smaller than 1, which specifies the confidence level for a signal and a stopping rule based on the comparison of the minimum deletion residual with a Bonferroni bound. For example if bonflev=0.99 the procedure stops when the trajectory exceeds for the first time the 99 per cent bonferroni bound.

2. a scalar value greater than 1. In this case the procedure stops when the residual trajectory exceeds for the first time this value.

Default value is empty, which means to rely on general rules based on consecutive exceedances.

It controls whether to display or not messages on the screen. If msg=TRUE (default) messages about the progression of the search are displayed on the screen otherwise only error messages will be displayed.

It controls whether to perform checks on matrix Y. If nocheck=TRUE, no check is performed.

Controls whether to monitor scaled Mahalanobis distances (only if monitoring=TRUE). If scaled=TRUE Mahalanobis distances monitored during the search are scaled using ratio of determinant. If scaled=2 Mahalanobis distances monitored during the search are scaled using asymptotic consistency factor. The default is scaled=FALSE, that is Mahalanobis distances are not scaled.

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

kx1 vector containing the list of the k units declared as outliers or NULL if the sample is homogeneous.

p-by-1 vector containing location of the data.

p-by-p robust estimate of covariance matrix.

n-by-1 vector containing the estimates of the robust Mahalanobis distances (in squared units). This vector contains the distances of each observation from the location of the data, relative to the scatter matrix cov.

(n-init)-by-2 matrix. 1st col is the forward search index; 2nd col is the value of minimum Mahalanobis Distance in each step of the fwd search.

(n-init)-by-11 matrix which contains the unit(s) included in the subset at each step of the fwd search. REMARK: in every step the new subset is compared with the old subset. Un contains the unit(s) present in the new subset but not in the old one. Un[1 ,2] for example contains the unit included in step init+1. Un[end, 2] contains the units included in the final step of the search.

2 x 5 matrix containing the number of times mdr went out of particular quantiles. First row contains quantiles 1 99 99.9 99.99 99.999. Second row contains the frequency distribution. It is NULL if bonflev threshold is used.

This output is produced only if the search found at a certain step is a non singular matrix X. In this case the search run in a constrained mode, that is including the units which produced a singular matrix in the last n-constr steps. out.constr is a vector which contains the list of units which produced a singular X matrix.

the data matrix X

Indicator for constant term. Scalar. If intercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

The number of observations that have determined the least trimmed squares estimator, scalar. h is an integer greater or equal than p but smaller then n. Generally if the purpose is outlier detection h=[0.5*(n+p+1)] (default value). h can be smaller than this threshold if the purpose is to find subgroups of homogeneous observations. In this function the LTS/LMS estimator is used just to initialize the search.

Number of subsamples which will be extracted to find the robust estimator, scalar. If nsamp=0 all subsets will be extracted. They will be (n choose p). If the number of all possible subset is <1000 the default is to extract all subsets otherwise just 1000.

Criterion to use to find the initial subset to initialize the search (LMS, LTS with concentration steps, LTS without concentration steps or subset supplied directly by the user). The default value is 1 (Least Median of Squares is computed to initialize the search). On the other hand, if the user wants to initialze the search with LTS with all the default options for concentration steps then lms=2. If the user wants to use LTS without concentration steps, lms can be a scalar different from 1 or 2. If lms is a list it is possible to control a series of options for concentration steps (for more details see option lms inside LXS_control). If, on the other hand, the user wants to initialize the search with a prespecified set of units there are two possibilities:

1. lms can be a vector with length greater than 1 which contains the list of units forming the initial subset. For example, if the user wants to initialize the search with units 4, 6 and 10 then lms=c(4, 6, 10);

2. lms is a struct which contains a field named bsb which contains the list of units to initialize the search. For example, in the case of simple regression through the origin with just one explanatory variable, if the user wants to initialize the search with unit 3 then lms=list(bsb=3).

Search initialization, scalar. It specifies the initial subset size to start monitoring exceedances of minimum deletion residual, if init is not specified it set equal to: p+1, if the sample size is smaller than 40 or min(3*p+1,floor(0.5*(n+p+1))), otherwise. For example, if init=100, the procedure starts monitoring from step m=100.

Check input arguments, scalar. If nocheck=TRUE no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added. As default nocheck=FALSE.

Option to be used if the distribution of the data is strongly non normal and, thus, the general signal detection rule based on consecutive exceedances cannot be used. In this case bonflev can be:

1. a scalar smaller than 1 which specifies the confidence level for a signal and a stopping rule based on the comparison of the minimum MD with a Bonferroni bound. For example if bonflev=0.99 the procedure stops when the trajectory exceeds for the first time the 99% bonferroni bound.

2. A scalar value greater than 1. In this case the procedure stops when the residual trajectory exceeds for the first time this value.

Default value is ”, which means to rely on general rules based on consecutive exceedances.

Controls whether to display or not messages on the screen If msg==1 (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

How to behave in case subset at step m (say bsbm) produces a singular X. In other words, this options controls what to do when rank(X[bsbm, ]) is smaller then number of explanatory variables. If bsbmfullrank=1 (default) these units (whose number is say mnofullrank) are constrained to enter the search in the final n-mnofullrank steps else the search continues using as estimate of beta at step m the estimate of beta found in the previous step.

Plot on the screen. Scalar. If plot=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. If plot=2 the user can also monitor the intermediate plots based on envelope superimposition. If plot=FALSE (default) no plot is produced.

Wheather to superimpose bivariate least square lines on the plot (if plot=TRUE. This option adds one or more least squares lines, based on SIMPLE REGRESSION of y on Xi, to the plots of y|Xi. The default is bivarfit=FALSE: no line is fitted. If bivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. If bivarfit=2, two OLS lines are fitted: one to all points and another to the group of the genuine observations. The group of the potential outliers is not fitted. If bivarfit=0 one OLS line is fitted to each group. This is useful for the purpose of fitting mixtures of regression lines. If bivarfit='i1' or bivarfit='i2', etc. an OLS line is fitted to a specific group, the one with index 'i' equal to 1, 2, 3 etc. Again, useful in case of mixtures.

Wheather to superimpose multivariate least square lines. This option adds one or more least square lines, based on MULTIVARIATE REGRESSION of y on X, to the plots of y|Xi. The default is multivarfit=FALSE: no line is fitted. If bivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. The line added to the scatter plot y|Xi is avconst + Ci*Xi, where Ci is the coefficient of Xi in the multivariate regression and avconst is the effect of all the other explanatory variables different from Xi evaluated at their centroid (that is overline(y)'C)). If multivarfit=2, same action as with multivarfit=1 but this time we also add the line based on the group of unselected observations (i.e. the normal units).

Add outlier labels in plot. If labeladd=TRUE, we label the outliers with the unit row index in matrices X and y. The default value is labeladd=FALSE, i.e. no label is added.

Add variable labels in plot. A vector of strings of length p containing the labels of the variables of the regression dataset. If it is empty (default) the sequence X1, ..., Xp will be created automatically

Add response label. A string containing the label of the response

Control y scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

Control x scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

p-by-1 vector containing the estimated regression parameters (in step n-k).

scalar containing the estimate of the scale (sigma).

residuals.

fitted values.

kx1 vector containing the list of the k units declared as outliers or NULL if the sample is homogeneous.

(n-init) x 2 matrix 1st col = fwd search index, 2nd col = value of minimum deletion residual in each step of the fwd search

(n-init) x 11 matrix which contains the unit(s) included in the subset at each step of the fwd search. REMARK: in every step the new subset is compared with the old subset. Un contains the unit(s) present in the new subset but not in the old one. Un(1,2) for example contains the unit included in step init+1. Un(end,2) contains the units included in the final step of the search.

2 x 5 matrix containing the number of times mdr went out of particular quantiles. First row contains quantiles 1 99 99.9 99.99 99.999. Second row contains the frequency distribution.

This output is produced only if the search found at a certain step is a non singular matrix X. In this case the search run in a constrained mode, that is including the units which produced a singular matrix in the last n-constr steps. out.constr is a vector which contains the list of units which produced a singular X matrix.

the data matrix X

the response vector y

a formula of the form y ~ x1 + x2 + ....

data frame from which variables specified in formula are to be taken.

an optional vector specifying a subset of observations to be used in the fitting process.

an optional vector of weights to be used in the fitting process. NOT USED YET.

a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The “factory-fresh” default is na.omit. Another possible value is NULL, no action. Value na.exclude can be useful.

logicals indicating if the model frame, the model matrix and the response are to be returned, respectively.

an optional list. See the contrasts.arg of model.matrix.default.

this can be used to specify an a priori known component to be included in the linear predictor during fitting. An offset term can be included in the formula instead or as well, and if both are specified their sum is used.

Predictor variables. Matrix. Matrix of explanatory variables (also called 'regressors') of dimension n x (p-1) where p denotes the number of explanatory variables including the intercept. Rows of X represent observations, and columns represent variables. By default, there is a constant term in the model, unless you explicitly remove it using input option intercept=FALSE, so do not include a column of 1s in X. Missing values (NA's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

Response variable. Vector. Response variable, specified as a vector of length n, where n is the number of observations. Each entry in y is the response for the corresponding row of X. Missing values (NA's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

Initial subset - vector of indices. If bsb=0 (default) then the procedure starts with p units randomly chosen. If bsb is not 0 the search will start with m0=length(bsb).

Indicator for constant term. Scalar. If intercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

wheather to perform monitoring for several quantities in each step of the forward search. Deafault is monitoring=FALSE.

A control object (S3) containing estimation options, as returned by FSR_control. Use the function FSR_control and see its help page. If the control object is supplied, the parameters from it will be used. If parameters are passed also in the invocation statement, they will override the corresponding elements of the control object.

Whether to print intermediate results. Default is trace=FALSE.

Potential further optional arguments, see the help of the function FSR_control.

Indicator for constant term. Scalar. If intercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Search initialization, scalar. It specifies the initial subset size to start monitoring exceedances of minimum deletion residual, if init is not specified it set equal to: p+1, if the sample size is smaller than 40 or min(3*p+1,floor(0.5*(n+p+1))), otherwise. For example, if init=100, the procedure starts monitoring from step m=100.

Check input arguments, scalar. If nocheck=TRUE no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added. As default nocheck=FALSE.

The kind of t-statistics which have to be monitored. tstat="trad" implies monitoring of traditional t statistics (out$Tols). In this case the estimate of s2 at step m is based on s2m (notice that s2m<<s2 when m/n is small) tstat="scal" (default) implies monitoring of rescaled t statistics. In this case the estimate of s2 at step m is based on s2m/vartruncnorm(m/n) where vartruncnorm(m/n) is the variance of the truncated normal distribution.

Confidence level which is used to declare units as outliers. Usually conflev=0.95, 0.975, 0.99 (individual alpha) or conflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975.

n x (n-init+1) matrix containing the monitoring of scaled residuals: the first row is the residual for the first unit, ..., n-th row is the residual for the n-th unit.

(n+1) x (n-init+1) matrix containing the monitoring of leverage: the first row is the leverage for the first unit, ..., n-th row is the leverage for the n-th unit.

n x (n-init+1) matrix containing the information about the units belonging to the subset at each step of the forward search: first col contains indexes of the units forming subset in the initial step; ...; last column contains units forming subset in the final step (all units).

n-init x 3 matrix which contains the monitoring of minimum deletion residual or (m+1)-ordered residual at each step of the forward search: first col is the fwd search index (from init to n-1); 2nd col = minimum deletion residual; 3rd col = (m+1)-ordered residual.

Remark: these quantities are stored with sign, that is the min deletion residual is stored with negative sign if it corresponds to a negative residual.

n-init+1 x 3 matrix which contains the monitoring of maximum studentized residual or m-th ordered residual: first col is the fwd search index (from init to n); 2nd col = maximum studentized residual; 3rd col = (m)-ordered studentized residual.

(n-init+1) x 4 matrix containing the monitoring of normality test in each step of the forward search: first col = fwd search index (from init to n); 2nd col = Asymmetry test; 3rd col = Kurtosis test; 4th col = Normality test.

(n-init+1) x (p+1) matrix containing the monitoring of estimated beta coefficients in each step of the forward search.

(n-init+1) x 5 matrix containing the monitoring of S2 or R2 and F-test in each step of the forward search:

1. 1st col = fwd search index (from init to n);

2. 2nd col = monitoring of S2;

3. 3rd col = monitoring of R2;

4. 4th col = monitoring of rescaled S2. In this case the estimated of sigma^2 at step m is divided by the consistency factor (to make the estimate asymptotically unbiased)

5. 5th col = monitoring of F test. Note that an asymptotic unbiased estimate of sigma^2 is used.

In this case the estimated of s2 at step m is divided by the consistency factor (to make the estimate asymptotically unbiased).

(n-init+1) x 3 matrix containing the monitoring of Cook or modified Cook distance in each step of the forward search:

1. 1st col = fwd search index (from init to n);

2. 2nd col = monitoring of Cook distance;

3. 3rd col = monitoring of modified Cook distance.

(n-init+1) x (p+1) matrix containing the monitoring of estimated t-statistics (as specified in option input 'tstat') in each step of the forward search

(n-init) x 11 matrix which contains the unit(s) included in the subset at each step of the fwd search.

REMARK: in every step the new subset is compared with the old subset. Un contains the unit(s) present in the new subset but not in the old one Un(1,2) for example contains the unit included in step init+1 Un(end,2) contains the units included in the final step of the search.

Confidence intervals for the elements of β. betaINT is a (n-init+1)-by-2*length(confint)-by-p 3D array. Each third dimension refers to an element of beta:

1. betaINT[,,1] is associated with first element of beta;

2. ...;

3. betaINT[,,p] is associated with last element of beta.

The first two columns contain the lower and upper confidence limits associated with conflev(1). Columns three and four contain the lower and upper confidence limits associated with conflev(2); ...; The last two columns contain the lower and upper confidence limits associated with conflev(end). For example betaINT[,3:4,5] contain the lower and upper confidence limits for the fifth element of beta using confidence level specified in the second element of input option conflev.

confidence interval for s2.

1. 1st col = fwd search index;

2. 2nd col = lower confidence limit based on conflev(1);

3. 3rd col = upper confidence limit based on conflev(1);

4. 4th col = lower confidence limit based on conflev(2);

5. 5th col = upper confidence limit based on conflev(2);

6. ...

7. penultimate col = lower confidence limit based on conflev(end);

8. last col = upper confidence limit based on conflev(end).

the data matrix X

the response vector y

a formula of the form y ~ x1 + x2 + ....

data frame from which variables specified in formula are to be taken.

an optional vector specifying a subset of observations to be used in the fitting process.

an optional vector of weights to be used in the fitting process. NOT USED YET.

a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The “factory-fresh” default is na.omit. Another possible value is NULL, no action. Value na.exclude can be useful.

logicals indicating if the model frame, the model matrix and the response are to be returned, respectively.

an optional list. See the contrasts.arg of model.matrix.default.

this can be used to specify an a priori known component to be included in the linear predictor during fitting. An offset term can be included in the formula instead or as well, and if both are specified their sum is used.

family of robust regression models, can be 'homo' for homoscedastic (same variance) regression model, 'hetero' for heteroskedastic regression model or 'bayes' Bayesian linear regression. The default is family='homo' for homoscedastic regression model.

robust regression estimation model, can be 'FS' for Forward search, 'S' for S regression, 'MM' for MM regression, 'LMS' or 'LTS'. The default is method='FS' for forward search estimation.

wheather to perform monitoring for several quantities in each step of the forward search or for series of values of the breakdown point in case of S estimates or for series of values of the efficiency in case of MM estimates. Deafault is monitoring=FALSE.

Response variable. Vector. Response variable, specified as a vector of length n, where n is the number of observations. Each entry in y is the response for the corresponding row of X. Missing values (NA's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

Predictor variables. Matrix. Matrix of explanatory variables (also called 'regressors') of dimension n x (p-1) where p denotes the number of explanatory variables including the intercept. Rows of X represent observations, and columns represent variables. By default, there is a constant term in the model, unless you explicitly remove it using input option intercept=FALSE, so do not include a column of 1s in X. Missing values (NA's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

Initial subset - vector of indices. If bsb=0 (default) then the procedure starts with p units randomly chosen. If bsb is not 0 the search will start with m0=length(bsb).

Indicator for constant term. Scalar. If intercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

A control object (S3) containing estimation options. If the control object is supplied, the parameters from it will be used. If parameters are passed also in the invocation statement, they will override the corresponding elements of the control object.

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

An n x p data matrix (n observations and p variables). Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

potential further arguments passed to lower level functions.

a formula of the form y ~ x1 + x2 + ....

data frame from which variables specified in formula are to be taken.

an optional vector specifying a subset of observations to be used in the fitting process.

an optional vector of weights to be used NOT USED YET.

a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The “factory-fresh” default is na.omit. Another possible value is NULL, no action. Value na.exclude can be useful.

logical indicating if the model frame, is to be returned.

logical indicating if the the model matrixis to be returned.

logical indicating if the response is to be returned.

an optional list. See the contrasts.arg of model.matrix.default.

this can be used to specify an a priori known component to be included in the linear predictor during fitting. An offset term can be included in the

Response variable. A vector with n elements that contains the response variable.

wheather to use constant term (default is intercept=TRUE

If plot=FALSE (default) no plot is produced. If plot=TRUE a fan plot is shown.

string which identifies the family of transformations which must be used. Possible values are c('BoxCox', 'YJ', 'YJpn', 'YJall'). Default is 'BoxCox'. The Box-Cox family of power transformations equals $(y^{\lambda}-1)/\lambda$ for $\lambda$ not equal to zero, and $\log(y)$ if $\lambda = 0$. The Yeo-Johnson (YJ) transformation is the Box-Cox transformation of $y+1$ for nonnegative values, and of $|y|+1$ with parameter $2-\lambda$ for $y$ negative. Remember that BoxCox can be used only if input y is positive. Yeo-Johnson family of transformations does not have this limitation. If family='YJpn' Yeo-Johnson family is applied but in this case it is also possible to monitor (in the output arguments Scorep and Scoren) the score test for positive and negative observations respectively. If family='YJall', it is also possible to monitor the joint F test for the presence of the two constructed variables for positive and negative observations.

values of the transformation parameter for which it is necessary to compute the score test. Default value of lambda is la=c(-1, -0.5, 0, 0.5, 1), i.e., the five most common values of lambda.

how to find the initlal subset to initialize the search. If lms=1 (default) Least Median of Squares (LMS) is computed, else Least Trimmed Squares (LTS) is computed. If, lms is matrix of size p - 1 + intercept X length(la) it contains in column j=1,..., lenght(la) the list of units forming the initial subset for the search associated with la(j). In this case the input option nsamp is ignored.

the percentage (roughly) of squared residuals whose sum will be minimized, by default alpha=0.5. In general, alpha must between 0.5 and 1.

The number of observations that have determined the least trimmed squares estimator, scalar. h is an integer greater or equal than p but smaller then n. Generally h=[0.5*(n+p+1)] (default value).

Search initialization. It specifies the initial subset size to start monitoring the value of the score test. If init is not specified it will be set equal to: p+1, if the sample size is smaller than 40 or min(3 * p + 1, floor(0.5 * (n+p+1))), otherwise.

Controls whether to display or not messages on the screen. If msg==TRUE messages are displayed on the screen. If msg=2, detailed messages are displayed, for example the information at iteration level.

Whether to check input arguments. If nocheck=TRUE no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added. The default is nocheck=FALSE.

number of subsamples which will be extracted to find the robust estimator. If nsamp=0 all subsets will be extracted. They will be n choose p.

Remark: if the number of all possible subset is <1000 the default is to extract all subsets otherwise just 1000. If nsamp is a matrix of size r-by-p, it contains in the rows the subsets which sill have to be extracted. For example, if p=3 and nsamp=c(2,4,9; 23, 45, 49; 90, 34, 1) the first subset is made up of units c(2, 4, 9), the second subset of units c(23, 45, 49) and the third subset of units c(90 34 1).

Confidence level for the bands (default is 0.99, that is, we plot two horizontal lines corresponding to values -2.58 and 2.58).

A label for the X-axis, default is 'Subset size m'

A label for the Y-axis, default is 'Score test statistic'

A label for the title, default is 'Fan plot'

Minimum and maximum for the X-axis

Minimum and maximum for the Y-axis

The line width of the curves which contain the score test, a positive number, default is lwd=2

The line width of the lines associated with the envelopes, a positive number, default is lwd.env=1

Whether to print intermediate results. Default is trace=FALSE.

a vector specifying the colors for the lines, each one corresponding to a la value. if length(col) < length(la), the colors will be recycled.

a vector specifying the line types for the lines, each one corresponding to a la value. if length(col) < length(la), the colors will be recycled.

An object containing monitoring of leverage, fsreda.object.

The needed elements of out are

1. LEV: matrix containing the leverage monitored in each step of the forward search. Every row is associated with a unit. This matrix can be created using function fsreg() with method="FS", monitoring=TRUE.

2. Un: (for FSR only) - matrix containing the order of entry in the subset of each unit (required only when datatooltip is true or databrush is not empty).

3. y: a vector containing the response (required only when option databrush is requested).

4. X: a matrix containing the explanatory variables (required only when option databrush is requested).

5. Bols: (n-init+1) x (p+1) matrix containing the estimated beta coefficients monitored in each step of the robust procedure (required only when option databrush is requested and suboption multivarfit is requested).

Control y scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

Control x scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

a title for the x axis

a title for the y axis

an overall title for the plot

The line width, a positive number, defaulting to 1

The line type. Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

colors to be used for the highlighted units

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

values for the x axis. Numeric vector of ncol(RES) controlling the x axis coordinates. The default value of xvalues is (nrow(RES) - ncol(RES) + 1):nrow(RES)

(alternative to fg.unit) numeric vector of length 1 or 2 which specifies the highlighted trajectories. If length(fthresh) == 1 the highlighted trajectories are those of units that throughtout the search had at leat once a residual greater (in absolute value) than thresh. The default value is fg.thresh=2.5. If length(fthresh) == 2 the highlighted trajectories are those of units that throughtout the search had a residual at leat once bigger than fg.thresh[2] or smaller than fg.thresh[1].

(alternative to fg.thresh), vector containing the list of the units to be highlighted. If fg.unit is supplied, fg.thresh is ignored.

numeric vector which specifies the steps of the search where to put labels for the highlighted trajectories (units). The default is to put the labels at the initial and final steps of the search. flabstep='' means no label.

The line width for the highlighted trajectories (units). Default is 1.

The line type for the highlighted trajectories (units). Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

colors to be used for the highlighted units.

Controlls whether to plot highlighted trajectories as symbols. if fg.mark==TRUE each line is plotted using a different symbol else no marker is used (default).

controls the font size of the labels of the trajectories in foreground.

numeric vector of length 1 or 2 which specifies how to define the unimmportant trajectories. Unimmportant trajectories will be plotted using a colormap, in greysh or will be hidden. If length(thresh) == 1 the irrelevant units are those which always had a residual smaller (in absolute value) than thresh. If length(bthresh) == 2 the irrelevant units are those which always had a residual greater than bthresh(1) and smaller than bthresh(2). The default is: bg.thresh=2.5 if n > 100 and bg.thresh=-Inf if n <= 100 i.e. to treat all trajectories as important if n <= 100 and, if n > 100, to reduce emphasis only to trajectories having in all steps of the search a value of scaled residual smaller than 2.5.

specifies how to plot the unimportant trajectories as defined in option bthresh.

1. bg.style="faint": unimportant trajectories are plotted using a colormap.

2. bg.style="hide": unimportant trajectories are hidden.

3. bg.style="greyish": unimportant trajectories are displayed in a faint grey.

When n>100 the default option is bg.style='faint'. When n <= 100 and bg.thresh == -Inf option bstyle is ignored. Remark: bground=” is equivalent to -Inf that is all trajectories are considered relevant.

Plot handle. String which identifies the handle of the plot which is about to be created. The default is to use tag 'pl_resfwd'. Notice that if the program finds a plot which has a tag equal to the one specified by the user, then the output of the new plot overwrites the existing one in the same window else a new window is created.

trajectories to highlight in connection with resfwdplot. If xground="lev" (default), the levfwdplot trajectories are put in foreground or in background depending on the leverage values. If xground="res", the levfwdplot trajectories are put in foreground or in background depending on the residual values. See options bg.thresh and fg.thresh.

Interactive clicking. It is inactive if this parameter is missing or empty. The default is datatooltip=TRUE, i.e. the user can select with the mouse an individual residual trajectory in order to have information about the corresponding unit. The information displayed depends on the estimator in use.

For example for class fsreda.object the information concerns the label and the step of the search in which the unit enters the subset. If datatooltip is a list it may contain the following fields:

1. DisplayStyle determines how the data cursor displays. Possible values are 'datatip' and 'window' (default). 'datatip' displays data cursor information in a small yellow text box attached to a black square marker at a data point you interactively select. 'window' displays data cursor information for the data point you interactively select in a floating window within the figure.

2. SnapToDataVertex: specifies whether the data cursor snaps to the nearest data value or is located at the actual pointer position. Possible values are SnapToDataVertex='on' (default) and SnapToDataVertex='off'.

3. LineColor: controls the color of the trajectory selected with the mouse. It can be an RGB triplet of values between 0 and 1, or character vector indicating a color name. Note that a RGB vector can be conveniently chosen with our MATLAB class FSColor, see documentation.

4. SubsetLinesColor: enables to control the color of the trajectories of the units that are in the subset at a given step of the search (if levfwdplot() is applied to an object of class fsreda.object) or have a weight greater than 0.9 (if levfwdplot() is applied to an object of class sregeda.object or mmregeda.object). This can be done (repeatedly) with a left mouse click in proximity of the step of interest. A right mouse click will terminate the selection by marking with a up-arrow the step corresponding to the highlighted lines. The highlighted lines by default are in red, but a different color can be specified as RGB triplet or character of color name. Note that a RGB vector can be conveniently chosen with our MATLAB class FSColor, see documentation. By default SubsetLinesColor="", i.e. the modality is not active. Any initialization for SubsetLinesColor which cannot be interpreted as RGB vector will be converted to blue, i.e. SubsetLinesColor will be forced to be [0 0 1]. If SubsetLinesColor is not empty the previous option LineColor is ignored.

Character vector containing the labels of the units (optional argument used when datatooltip=TRUE. If this field is not present labels row1, ..., rown will be automatically created and included in the pop up datatooltip window).

Add variable labels in plot. A vector of strings of length p containing the labels of the variables of the regression dataset. If it is empty (default) the sequence X1, ..., Xp will be created automatically

Add response label. A string containing the label of the response

Controls whether to display or not messages on the screen If msg==1 (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

interactive mouse brushing. If databrush is missing or empty (default), no brushing is done. The activation of this option (databrush is a scalar or a list) enables the user to select a set of trajectories in the current plot and to see them highlighted in the y|X plot, i.e. a matrix of scatter plots of y against each column of X, grouped according to the selection(s) done by brushing. If the plot y|X does not exist it is automatically created. In addition, brushed units are automatically highlighted in the minimum deletion residual plot if it is already open. The extension to the following plots will be available in future versions of the toolbox:

1. monitoring leverage plot;

2. maximum studentized residual;

3. s^2 and R^2;

4. Cook distance and modified Cook distance;

5. deletion t statistics.

Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too

If databrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

If databrush=list(...), it is possible to use all optional arguments of function selectdataFS() and the following optional argument:

1. persist. Persist is an empty value or a character containing 'on' or 'off'. The default value is persist="", that is brushing is allowed only once. If persist="on" or persis="off" brushing can be done as many time as the user requires. If persist='on' then the unit(s) currently brushed are added to those previously brushed. It is possible, every time a new brushing is done, to use a different color for the brushed units. If persist='off' every time a new brush is performed units previously brushed are removed.

2. bivarfit. Wheather to superimpose bivariate least square lines on the plot (if plot=TRUE. This option adds one or more least squares lines, based on SIMPLE REGRESSION of y on Xi, to the plots of y|Xi. The default is bivarfit=FALSE: no line is fitted. If bivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. If bivarfit=2, two OLS lines are fitted: one to all points and another to the group of the genuine observations. The group of the potential outliers is not fitted. If bivarfit=0 one OLS line is fitted to each group. This is useful for the purpose of fitting mixtures of regression lines. If bivarfit='i1' or bivarfit='i2', etc. an OLS line is fitted to a specific group, the one with index 'i' equal to 1, 2, 3 etc. Again, useful in case of mixtures.

3. multivarfit. Wheather to superimpose multivariate least square lines. This option adds one or more least square lines, based on MULTIVARIATE REGRESSION of y on X, to the plots of y|Xi. The default is multivarfit=FALSE: no line is fitted. If bivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. The line added to the scatter plot y|Xi is avconst + Ci*Xi, where Ci is the coefficient of Xi in the multivariate regression and avconst is the effect of all the other explanatory variables different from Xi evaluated at their centroid (that is overline(y)'C)). If multivarfit=2, same action as with multivarfit=1 but this time we also add the line based on the group of unselected observations (i.e. the normal units).

4. labeladd. Add outlier labels in plot. If labeladd=TRUE, we label the outliers with the unit row index in matrices X and y. The default value is labeladd=FALSE, i.e. no label is added.

(MATLAB-style arguments) appearance of the plot in terms of xlim, ylim, axes labels and their font size style, color of the lines, etc.

MATLAB-style arguments for the fground trajectories in foregroud.

MATLAB-style arguments for the fground trajectories in backgroud.

potential further arguments passed to lower level functions.

Indicator for constant term. Scalar. If intercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Criterion to use to find the initial subset to initialize the search (LMS, LTS with concentration steps, LTS without concentration steps or subset supplied directly by the user). The default value is 1 (Least Median of Squares is computed to initialize the search). On the other hand, if the user wants to initialze the search with LTS with all the default options for concentration steps then lms=2. If the user wants to use LTS without concentration steps, lms can be a scalar different from 1 or 2. If lms is a list it is possible to control a series of options for concentration steps (for more details see option lms inside LXS_control). If, on the other hand, the user wants to initialize the search with a prespecified set of units there are two possibilities:

1. lms can be a vector with length greater than 1 which contains the list of units forming the initial subset. For example, if the user wants to initialize the search with units 4, 6 and 10 then lms=c(4, 6, 10);

2. lms is a struct which contains a field named bsb which contains the list of units to initialize the search. For example, in the case of simple regression through the origin with just one explanatory variable, if the user wants to initialize the search with unit 3 then lms=list(bsb=3).

The number of observations that have determined the least trimmed squares estimator, scalar. h is an integer greater or equal than p but smaller then n. Generally if the purpose is outlier detection h=[0.5*(n+p+1)] (default value). h can be smaller than this threshold if the purpose is to find subgroups of homogeneous observations. In this function the LTS/LMS estimator is used just to initialize the search.

Breakdown point. It measures the fraction of outliers the algorithm should resist. In this case any value greater than 0 but smaller or equal than 0.5 will do fine. If on the other hand the purpose is subgroups detection then bdp can be greater than 0.5. In any case however n*(1-bdp) must be greater than p. If this condition is not fulfilled an error will be given. Please specify h or bdp not both.

Number of subsamples which will be extracted to find the robust estimator, scalar. If nsamp=0 all subsets will be extracted. They will be (n choose p). If the number of all possible subset is <1000 the default is to extract all subsets otherwise just 1000.

LXS reweighted - if rew=1 the reweighted version of LTS (LMS) is used and the output quantities refer to the reweighted version else no reweighting is performed (default).

Confidence level which is used to declare units as outliers, usually conflev=0.95, 0.975, 0.99 (individual alpha) or 1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975.

Controls whether to display or not messages on the screen If msg==1 (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

Check input arguments, scalar. If nocheck=TRUE no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added. As default nocheck=FALSE.

It controls whether to display or not on the screen messages about estimated time to compute LMS (LTS). If nomes is equal to 1 no message about estimated time to compute LMS (LTS) is displayed, else if nomes is equal to 0 (default), a message about estimated time is displayed.

Plot on the screen. Scalar. If plots=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. If plots=2 the user can also monitor the intermediate plots based on envelope superimposition. If plots=FALSE (default) no plot is produced.

An object of S3 class fsmeda.object returned by fsmult with monitoring=TRUE - a list containing the monitoring of minimum Mahalanobis distance.

The needed elements of out are

1. MAL: matrix containing the squared Mahalanobis distances monitored in each step of the forward search. Every row is associated with a unit (row of data matrix X).

2. Un: matrix containing the order of entry of each unit (necessary if datatooltip or databrush is selected).

3. X: The data matrix.

Controls the x scale in the plot. xlim is a vector with two elements controlling minimum and maximum on the x-axis. Default is to use automatic scale.

Controls the y scale in the plot. ylim is a vector with two elements controlling minimum and maximum on the y-axis. Default is to use automatic scale.

A title for the x axis

A title for the y axis, deafult is "Squared Mahalanobis distances".

An overall title for the plot

The line width, a positive number, defaulting to 1

The line type. Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

Colors to be used for the highlighted units

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Numeric vector containing the subset size with length equal to the number of columns of matrix of mahalanobis distances. The default value of subsize is (nrow(MAL) - ncol(MAL) + 1):nrow(MAL)

(alternative to fg.unit) numeric vector of length 1 or 2 which specifies the highlighted trajectories. If length(fg.thresh) == 1 the highlighted trajectories are those of units that throughtout the search had at leat once a mahalanobis distance greater than fg.thresh. The default value is fg.thresh=2.5. If length(fg.thresh) == 2 the highlighted trajectories are those of units that throughtout the search had a mahalanobis distance at least once bigger than fg.thresh[2] or smaller than fg.thresh[1].

(alternative to fg.thresh), vector containing the list of the units to be highlighted. If fg.unit is supplied, fg.thresh is ignored.

numeric vector which specifies the steps of the search where to put labels for the highlighted trajectories (units). The default is to put the labels at the initial and final steps of the search. fg.labstep='' means no label.

The line width for the highlighted trajectories (units). Default is 1.

The line type for the highlighted trajectories (units). Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

colors to be used for the highlighted units.

Controlls whether to plot highlighted trajectories as symbols. if fg.mark==TRUE each line is plotted using a different symbol else no marker is used (default).

Controls the font size of the labels of the trajectories in foreground. If fg.cex=0 no labels will be shown - equivalent to fg.labstop="".

Numeric vector of length 1 or 2 which specifies how to define the unimmportant trajectories. Unimmportant trajectories will be plotted using a colormap, in greysh or will be hidden. If length(bg.thresh) == 1 the irrelevant units are those which always had a mahalanobis distance smaller than bg.thresh. If length(bg.thresh) == 2 the irrelevant units are those which always had a mahalanobis distance greater than bg.thresh[1] and smaller than bg.thresh[2]. The default is bg.thresh=2.5 if n > 100 and bg.thresh=-Inf if n <= 100 i.e. to treat all trajectories as important if n <= 100 and, if n > 100, to reduce emphasis only to trajectories having in all steps of the search a value of mahalanobis distance smaller than 2.5.

Specifies how to plot the unimportant trajectories as defined in option bg.thresh.

1. bg.style="faint": unimportant trajectories are plotted using a colormap.

2. bg.style="hide": unimportant trajectories are hidden.

3. bg.style="greyish": unimportant trajectories are displayed in a faint grey.

When n > 100 the default option is bg.style='faint'. When n <= 100 and bg.thresh == -Inf option bg.style is ignored. Remark: bground=” is equivalent to -Inf that is all trajectories are considered relevant.

MATLAB-style arguments - appearance of the plot in terms of xlim, ylim, axes labels and their font size style, color of the lines, etc.

MATLAB-style arguments - for the trajectories in foregroud.

MATLAB-style arguments - for the trajectories in backgroud.

Plot handle. String which identifies the handle of the plot which is about to be created. The default is to use tag 'pl_resfwd'. Notice that if the program finds a plot which has a tag equal to the one specified by the user, then the output of the new plot overwrites the existing one in the same window else a new window is created.

Interactive clicking. It is inactive if this parameter is set to FALSE. The default is datatooltip=TRUE, the user can select with the mouse an individual mahalanobis distance trajectory in order to have information about the corresponding unit, the associated label and the step of the search in which the unit enters the subset. If datatooltip is a list it may contain the following fields:

1. DisplayStyle determines how the data cursor displays. Possible values are 'datatip' and 'window' (default). 'datatip' displays data cursor information in a small yellow text box attached to a black square marker at a data point you interactively select. 'window' displays data cursor information for the data point you interactively select in a floating window within the figure.

2. SnapToDataVertex: specifies whether the data cursor snaps to the nearest data value or is located at the actual pointer position. Possible values are SnapToDataVertex='on' (default) and SnapToDataVertex='off'.

3. LineColor: controls the color of the trajectory selected with the mouse. It can be an RGB triplet of values between 0 and 1, or character vector indicating a color name. Note that a RGB vector can be conveniently chosen with our MATLAB class FSColor, see documentation.

4. SubsetLinesColor: enables to control the color of the trajectories of the units that are in the subset at a given step of the search (if resfwdplot() is applied to an object of class fsreda.object) or have a weight greater than 0.9 (if resfwdplot() is applied to an object of class sregeda.object or mmregeda.object). This can be done (repeatedly) with a left mouse click in proximity of the step of interest. A right mouse click will terminate the selection by marking with a up-arrow the step corresponding to the highlighted lines. The highlighted lines by default are in red, but a different color can be specified as RGB triplet or character of color name. Note that a RGB vector can be conveniently chosen with our MATLAB class FSColor, see documentation. By default SubsetLinesColor="", i.e. the modality is not active. Any initialization for SubsetLinesColor which cannot be interpreted as RGB vector will be converted to blue, i.e. SubsetLinesColor will be forced to be [0 0 1]. If SubsetLinesColor is not empty the previous option LineColor is ignored.

Character vector containing the labels of the units (optional argument used when datatooltip=TRUE. If this field is not present labels row1, ..., rown will be automatically created and included in the pop up datatooltip window).

Add variable labels in plot. A vector of strings of length p containing the labels of the variables in the dataset. If it is empty (default) the sequence X1, ..., Xp will be created automatically

Interactive mouse brushing. If databrush is missing or empty (default), no brushing is done. The activation of this option (databrush is TRUE or a list) enables the user to select a set of trajectories in the current plot and to see them highlighted in the scatterplot matrix. If the scatterplot matrix does not exist it is automatically created. In addition, brushed units can be highlighted in the monitoring mahalanobis distance plot. Note that the window style of the other figures is set equal to that which contains the monitoring mahalanobis distance plot. In other words, if the monitoring mahalanobis distance plot is docked all the other figures will be docked too.

If databrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

If databrush=list(...), it is possible to use all optional arguments of the MATLAB function selectdataFS() and the following optional arguments:

• persist. Persist is an empty value or a character containing 'on' or 'off'. The default value is persist="", that is brushing is allowed only once. If persist="on" or persis="off" brushing can be done as many time as the user requires. If persist='on' then the unit(s) currently brushed are added to those previously brushed. It is possible, every time a new brushing is done, to use a different color for the brushed units. If persist='off' every time a new brush is performed units previously brushed are removed.

• label: add labels of brushed units in the monitoring plot.

• labeladd: add labels of brushed units in the scatterplot matrix. If this option is '1', we label the units of the last selected group with the unit row index in the matrix X. The default value is labeladd=”, i.e. no label is added.

confidence interval for the horizontal bands. It can be a vector of different confidence level values, e.g. c(0.95, 0.99, 0.999). The confidence interval is based on the $chi^2$ distribution.

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

a numeric vector or an object of S3 class (one of fsmult.object, smult.object or mmmult.object) returned by one of the functions fsmult or smult or mmmult - a list containing the monitoring of minimum Mahalanobis distance

If out is a vector, p is the number of variables of the original data matrix which have been used to compute md.

A title for the x axis

A title for the y axis

An overall title for the plot

Add variable labels in the plot. A vector of strings of length p containing the labels of the variables of the original data matrix X. If it is empty (default) the sequence X1, ..., Xp will be created automatically

confidence interval for the horizontal bands. It can be a vector of different confidence level values, e.g. c(0.95, 0.99, 0.999). The confidence interval is based on the chi^2 distribution.

Number of points to be labeled in the plot. If numlab is a single number, e.g. numlab]k, the units with the k largest md are labelled in the plots. If numlab is a vector, the units indexed by the vector are labelled in the plot. Default is numlab=5, i.e. the 5 units units with the largest md are labelled. Use numlab="" for no labelling.

Tag of the figure which will host the malindexplot. The default tag is tag="pl_malindex".

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

An object returned by FSReda() (see FSReda_control).

The needed elements of out are

1. mdr: Minimum deletion residual. A matrix containing the monitoring of minimum deletion residual in each step of the forward search. The first column of mdr must contain the fwd search index.

2. Un: (for FSR only) - matrix containing the order of entry in the subset of each unit (required only when datatooltip is true or databrush is not empty).

3. y: a vector containing the response (required only when option databrush is requested).

4. X: a matrix containing the explanatory variables (required only when option databrush is requested).

5. Bols: (n-init+1) x (p+1) matrix containing the estimated beta coefficients monitored in each step of the robust procedure (required only when option databrush is requested and suboption multivarfit is requested).

Quantiles for which envelopes have to be computed. The default is to produce 1%, 50% and 99% envelopes. In other words the default is quant=c(0.01, 0.5, 0.99).

Wheather to use MDR with sign: if sign=TRUE (default) we distinguish steps for which minimum deletion residual was associated with positive or negative value of the residual. Steps associated with positive values of mdr are plotted in black, while other steps are plotted in red.

Wheather to plot the (m+1)-th order statistic. Specifies if it is necessary to plot the curve associated with (m+1)-th order statistic.

Sample size for drawing enevlopes. Specifies the size of the sample which is used to superimpose the envelope. The default is to add an envelope based on all the observations (size n envelope).

Control y scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

Control x scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

a title for the x axis

a title for the y axis

an overall title for the plot

Controls the width of the lines associated with the envelopes, default is lvdenv=1.

Controls the linewidth of the curve which contains the monitoring of minimum deletion residual.

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Plot handle. String which identifies the handle of the plot which is about to be created. The default is to use tag 'pl_mdr'. Notice that if the program finds a plot which has a tag equal to the one specified by the user, then the output of the new plot overwrites the existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to have information about the unit selected, the step in which the unit enters the search and the associated label. If datatooltip is a list, it is possible to control the aspect of the data cursor (see MATLAB function datacursormode() for more details or see the examples below). The default options are DisplayStyle="Window" and SnapToDataVertex="on".

Character vector containing the labels of the units (optional argument used when datatooltip=TRUE. If this field is not present labels row1, ..., rown will be automatically created and included in the pop up datatooltip window).

Add variable labels in plot. A vector of strings of length p containing the labels of the variables of the regression dataset. If it is empty (default) the sequence X1, ..., Xp will be created automatically

Add response label. A string containing the label of the response

interactive mouse brushing. If databrush is missing or empty (default), no brushing is done. The activation of this option (databrush is a scalar or a list) enables the user to select a set of trajectories in the current plot and to see them highlighted in the y|X plot, i.e. a matrix of scatter plots of y against each column of X, grouped according to the selection(s) done by brushing. If the plot y|X does not exist it is automatically created. In addition, brushed units are automatically highlighted in the minimum deletion residual plot if it is already open. The extension to the following plots will be available in future versions of the toolbox:

1. monitoring leverage plot;

2. maximum studentized residual;

3. s^2 and R^2;

4. Cook distance and modified Cook distance;

5. deletion t statistics.

Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too

If databrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

If databrush=list(...), it is possible to use all optional arguments of function selectdataFS() and the following optional argument:

1. persist. Persist is an empty value or a character containing 'on' or 'off'. The default value is persist="", that is brushing is allowed only once. If persist="on" or persis="off" brushing can be done as many time as the user requires. If persist='on' then the unit(s) currently brushed are added to those previously brushed. It is possible, every time a new brushing is done, to use a different color for the brushed units. If persist='off' every time a new brush is performed units previously brushed are removed.

2. bivarfit. Wheather to superimpose bivariate least square lines on the plot (if plot=TRUE. This option adds one or more least squares lines, based on SIMPLE REGRESSION of y on Xi, to the plots of y|Xi. The default is bivarfit=FALSE: no line is fitted. If bivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. If bivarfit=2, two OLS lines are fitted: one to all points and another to the group of the genuine observations. The group of the potential outliers is not fitted. If bivarfit=0 one OLS line is fitted to each group. This is useful for the purpose of fitting mixtures of regression lines. If bivarfit='i1' or bivarfit='i2', etc. an OLS line is fitted to a specific group, the one with index 'i' equal to 1, 2, 3 etc. Again, useful in case of mixtures.

3. multivarfit. Wheather to superimpose multivariate least square lines. This option adds one or more least square lines, based on MULTIVARIATE REGRESSION of y on X, to the plots of y|Xi. The default is multivarfit=FALSE: no line is fitted. If bivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. The line added to the scatter plot y|Xi is avconst + Ci*Xi, where Ci is the coefficient of Xi in the multivariate regression and avconst is the effect of all the other explanatory variables different from Xi evaluated at their centroid (that is overline(y)'C)). If multivarfit=2, same action as with multivarfit=1 but this time we also add the line based on the group of unselected observations (i.e. the normal units).

4. labeladd. Add outlier labels in plot. If labeladd=TRUE, we label the outliers with the unit row index in matrices X and y. The default value is labeladd=FALSE, i.e. no label is added.

potential further arguments passed to lower level functions.

An object of S3 class fsmeda.object returned by fsmult with monitoring=TRUE - a list containing the monitoring of minimum Mahalanobis distance

Quantiles for which envelopes have to be computed. The default is to produce 1%, 50% and 99% envelopes. In other words the default is quant=c(0.01, 0.5, 0.99).

Wheather to plot the (m+1)-th order statistic.

Sample size for drawing enevlopes. Specifies the size of the sample which is used to superimpose the envelope. The default is to add an envelope based on all the observations (size n envelope).

Controls the line width of the curve which contains the monitoring of minimum deletion residual.

Controls the width of the lines associated with the envelopes. Default is lwdenv=1

Control the x scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

Control the y scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

Plot handle. String which identifies the handle of the plot which is about to be created. The default is tag='pl_mmd'. Notice that if the program finds a plot which has a tag equal to the one specified by the user, then the output of the new plot overwrites the existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to have information about the unit selected, the step in which the unit enters the search and the associated label. If datatooltip is a list, it is possible to control the aspect of the data cursor (see MATLAB function datacursormode() for more details or see the examples below). The default options are DisplayStyle="Window" and SnapToDataVertex="on".

Row labels. Character vector containing the labels of the units (optional argument used when datatooltip=TRUE. If this field is not present labels row1, ..., rown will be automatically created and included in the pop up datatooltip window).

A title for the x axis

A title for the y axis

An overall title for the plot

Add variable labels in the plot. A vector of strings of length p containing the labels of the variables of the original data matrix X. If it is empty (default) the sequence X1, ..., Xp will be created automatically

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Interactive mouse brushing. If databrush is missing or empty (default), no brushing is done. The activation of this option (databrush is TRUE or a list) enables the user to select a set of trajectories in the current plot and to see them highlighted in the scatterplot matrix. If the scatterplot matrix does not exist it is automatically created. In addition, brushed units can be highlighted in the monitoring MD plot. Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too.

If databrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too

If databrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

If databrush=list(...), it is possible to use all optional arguments of the MATLAB function selectdataFS() and the following optional arguments:

• persist: This option can be an empty value or a character containing 'on' or 'off'. The default value is persist="", that is brushing is allowed only once. If persist="on" or persis="off" brushing can be done as many time as the user requires. If persist='on' then the unit(s) currently brushed are added to those previously brushed. It is possible, every time a new brushing is done, to use a different color for the brushed units. If persist='off' every time a new brush is performed units previously brushed are removed.

• labeladd: add labels of brushed units in the scatterplot matrix. If this option is '1', we label the units of the last selected group with the unit row index in the matrix X. The default value is labeladd=”, i.e. no label is added.

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

An object of S3 class fsmmmdrs.object returned by fsmmmdrs - a list containing the following elements:

• mmdrs = a matrix of size (n-ninit)-by-(nsimul+1) containing the monitoring of minimum Mahalanobis distance in each step of the forward search for each of the nsimul random starts. The first column of mmdrs must contain the forward search index. This matrix can be created using function fsmmmdrs.

• BBrs = 3D array of size n-by-n-(init)-by-nsimul containing units forming subset for rach random start. This field is necessary if datatooltip is true or databrush is not empty.

• X = n-by-v matrix containing the original data matrix. This field is necessary if datatooltip is true or databrush is not empty.

Quantiles for which envelopes have to be computed. The default is to produce 1%, 50% and 99% envelopes. In other words the default is quant=c(0.01, 0.5, 0.99).

Sample size for drawing enevlopes. Specifies the size of the sample which is used to superimpose the envelope. The default is to add an envelope based on all the observations (size n envelope).

Controls the linewidth of the curve which contains the monitoring of minimum deletion residual.

line width: a scalar which controls the width of the lines associated with the envelopes. Default is lwdenv=1

Control the x scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

Control the y scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

Plot handle. String which identifies the handle of the plot which is about to be created. The default is tag='pl_mmdrs'. Notice that if the program finds a plot which has a tag equal to the one specified by the user, then the output of the new plot overwrites the existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to have information about the unit selected, the step in which the unit enters the search and the associated label. If datatooltip is a list, it is possible to control the aspect of the data cursor (see MATLAB function datacursormode() for more details or see the examples below). The default options are DisplayStyle="Window" and SnapToDataVertex="on".

Row labels. Character vector containing the labels of the units (optional argument used when datatooltip=TRUE. If this field is not present labels row1, ..., rown will be automatically created and included in the pop up datatooltip window).

A title for the x axis

A title for the y axis

wheather to label the envelopes. If envlab is true (default) labels of the confidence envelopes which are used are added on the y axis.

An overall title for the plot

Add variable labels in the plot. A vector of strings of length p containing the labels of the variables of the original data matrix X. If it is empty (default) the sequence X1, ..., Xp will be created automatically

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Interactive mouse brushing. If databrush is missing or empty (default), no brushing is done. The activation of this option (databrush is TRUE or a list) enables the user to select a set of trajectories in the current plot and to see them highlighted in the scatterplot matrix. If the scatterplot matrix does not exist it is automatically created. In addition, brushed units can be highlighted in the monitoring MD plot. Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too.

If databrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too

If databrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

If databrush=list(...), it is possible to use all optional arguments of the MATLAB function selectdataFS() and the following optional arguments:

1. persist. Persist is an empty value or a character containing 'on' or 'off'. The default value is persist="", that is brushing is allowed only once. If persist="on" or persis="off" brushing can be done as many time as the user requires. If persist='on' then the unit(s) currently brushed are added to those previously brushed. It is possible, every time a new brushing is done, to use a different color for the brushed units. If persist='off' every time a new brush is performed units previously brushed are removed.

Wheather to use scaled or unscaled envelopes. If scaled=TRUE the envelopes are produced for scaled Mahalanobis distances (no consistency factor is applied) else the traditional consistency factor is applied. Default is scaled=FALSE

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

An n x p data matrix (n observations and p variables). Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

Wheather to perform monitoring of Mahalanobis distances and other specific quantities

Plots the Mahalanobis distances against index number. If plot=FALSE (default) or plot=0 no plot is produced. The confidence level used to draw the confidence bands for the MD is given by the input option conflev. If conflev is not specified a nominal 0.975 confidence interval will be used. If plot=2 a scatter plot matrix with the outliers highlighted is produced. If plot is a list it may contain the following fields:

• labeladd If labeladd=1, the outliers in the spm are labelled with the unit row index. The default value is labeladd="", i.e. no label is added

• nameY character vector containing the labels of the variables. As default value, the labels which are added are Y1, ...Yp.

Defining the nominal efficiency (i.e. a number between 0.5 and 0.99). The default value is eff=0.95.

Confidence level which is used to declare units as outliers (scalar). Usually conflev=0.95, conflev=0.975 or conflev=0.99 (individual alpha) conflev=1-0.05/n, conflev=1-0.025/n or conflev=1-0.01/n (simultaneous alpha). Default value is convlev=0.975.

It controls whether to perform checks on matrix Y. If nocheck=TRUE, no check is performed.

Whether to print intermediate results. Default is trace=FALSE.

potential further arguments passed to lower level functions.

p-by-1 vector containing MM estimate of location.

p-by-p matrix with MM estimate of the shape matrix.

matrix with MM estimate of the covariance matrix. Remark: covariance = auxscale^2 * shape.

A vector containing the estimates of the weights.

A vector containing the list of the units declared as outliers using confidence level specified in input scalar conflev.

A vector with S estimate of location.

A matrix with S estimate of the shape matrix.

A matrix with S estimate of the covariance matrix.

S estimate of the scale.

n-by-1 vector containing the estimates of the robust Mahalanobis distances (in squared units).

Confidence level that was used to declare outliers.

the data matrix X

length(eff)-by-p matrix containing MM estimate of location for each value of eff.

p-by-p-by-length(eff) 3D array containing robust estimate of the shape for each value of eff. Remark: det|shape|=1.

length(eff) vector containing robust estimate of the scale for each value of eff.

p-by-p-by-length(eff) 3D array containing robust estimate of covariance matrix for each value of eff. Note that scale(i)^2 * shape[,,i] = robust estimate of covariance matrix.

(p+1)-by-length(eff) matrix containing the units forming best subset for each value of eff.

n-by-length(eff) matrix containing the estimates of the robust Mahalanobis distances (in squared units) for each value of eff.

n-by-length(eff) matrix containing flags for the outliers. Boolean matrix containing the list of the units declared as outliers for each value of eff using confidence level specified in input scalar conflev

n x length(eff) matrix containing the weights for each value of eff.

Confidence level that was used to declare outliers.

Number of subsets without full rank. Notice that singsub > 0.1*(number of subsamples) produces a warning.

vector which contains the values of eff which have been used.

the data matrix X.

Indicator for constant term. Scalar. If intercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Starting values of the MM-estimator, a list with the fiollowing elements: loc, a $p x 1$ vector, location vector estimate and scale, a scaler, estimate of the scale. If empty (default) the program will use S estimators. In this last case it is possible to specify the options given in function Sreg.

Scalar defining nominal efficiency (i.e. a number between 0.5 and 0.99). The default value is 0.95.

Location or scale efficiency. If effshape=1 efficiency refers to shape efficiency else (default) efficiency refers to location efficiency.

Specifies the rho function which must be used to weight the residuals. Possible values are 'bisquare' 'optimal' 'hyperbolic' 'hampel'.

1. 'bisquare' uses Tukey's rho and psi functions. See TBrho and TBpsi.

2. 'optimal' uses optimal rho and psi functions. See OPTrho and OPTpsi.

3. 'hyperbolic' uses hyperbolic rho and psi functions. See HYPrho and HYPpsi.

4. 'hampel' uses Hampel rho and psi functions. See HArho and HApsi.

The default is 'bisquare'.

Additional parameters for the specified rho function. For hyperbolic rho function it is possible to set up the value of k = sup CVC (the default value of k is 4.5).

For Hampel rho function it is possible to define parameters a, b and c (the default values are a=2, b=4, c=8)

Number of refining iterations in each subsample (default is refsteps=3). refsteps = 0 means "raw-subsampling" without iterations.

Scalar controlling tolerance in the MM loop. The default value is tol=1e-6

Confidence level which is used to declare units as outliers. Usually conflev=0.95, 0.975, 0.99 (individual alpha) or conflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975

Check input arguments, scalar. If nocheck=TRUE no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added. As default nocheck=FALSE.

Controls whether to display or not messages on the screen If Smsg==TRUE (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

Plot on the screen. Scalar. If plots=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. If plots=2 the user can also monitor the intermediate plots based on envelope superimposition. If plots=FALSE (default) no plot is produced.

p-by-1 vector containing the MM estimate of regression coefficients.

scalar, S estimate of the scale (or supplied external estimate of scale, if option InitialEst is not empty).

residuals.

fitted values.

n x 1 vector. Weights assigned to each observation.

p x 1 vector containing S estimate of regression coefficients (or supplied initial external estimate of regression coefficients, if option InitialEst is not empty)

Number of subsets without full rank in the S preliminary part. Notice that out.singsub > 0.1*(number of subsamples) produces a warning.

kx1 vector containing the list of the k units declared as outliers or NULL if the sample is homogeneous.

Confidence level which is used to declare units as outliers. Usually conflev=0.95, 0.975, 0.99 (individual alpha) or conflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975

Specifies the rho function which has been used to weight the residuals. If a different rho function is specified for S and MM loop then insted of rhofunc we will have rhofuncS and rhofuncMM.

Vector which contains the additional parameters for the specified rho function which has been used. For hyperbolic rho function the value of k =sup CVC. For Hampel rho function the parameters a, b and c. If a different rho function is specified for S and MM loop then insted of rhofuncparam we will have rhofuncparamS and rhofuncparamMM.

the data matrix X

the response vector y

Indicator for constant term. Scalar. If intercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Starting values of the MM-estimator, a list with the fiollowing elements: loc, a $p x 1$ vector, location vector estimate and scale, a scaler, estimate of the scale. If empty (default) the program will use S estimators. In this last case it is possible to specify the options given in function Sreg.

Options to pass to Sreg, an Sreg_control object. The options are: Srhofunc, Snsamp, Srefsteps, Sreftol, Srefstepsbestr, Sreftolbestr, Sminsctol, Sbestr. See function Sreg_control for more details on these options.

It is necessary to add to the S options the letter S at the beginning. For example, if you want to use the optimal rho function the supplied option is 'Srhofunc','optimal'. For example, if you want to use 3000 subsets, the supplied option is 'Snsamp',3000

Vector defining nominal efficiency (i.e. a series of numbers between 0.5 and 0.99). The default value is the sequence seq(0.5, 0.99, 0.01)

Location or scale efficiency. If effshape=1 efficiency refers to shape efficiency else (default) efficiency refers to location efficiency.

Specifies the rho function which must be used to weight the residuals. Possible values are 'bisquare' 'optimal' 'hyperbolic' 'hampel'.

1. 'bisquare' uses Tukey's rho and psi functions. See TBrho and TBpsi.

2. 'optimal' uses optimal rho and psi functions. See OPTrho and OPTpsi.

3. 'hyperbolic' uses hyperbolic rho and psi functions. See HYPrho and HYPpsi.

4. 'hampel' uses Hampel rho and psi functions. See HArho and HApsi.

The default is 'bisquare'.

Additional parameters for the specified rho function. For hyperbolic rho function it is possible to set up the value of k = sup CVC (the default value of k is 4.5).

For Hampel rho function it is possible to define parameters a, b and c (the default values are a=2, b=4, c=8)

Number of refining iterations in each subsample (default is refsteps=3). refsteps = 0 means "raw-subsampling" without iterations.

Scalar controlling tolerance in the MM loop. The default value is tol=1e-6.

Confidence level which is used to declare units as outliers. Usually conflev=0.95, 0.975, 0.99 (individual alpha) or conflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975

Check input arguments, scalar. If nocheck=TRUE no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added. As default nocheck=FALSE.

Plot on the screen. Scalar. If plots=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. If plots=2 the user can also monitor the intermediate plots based on envelope superimposition. If plots=FALSE (default) no plot is produced.

scalar, S estimate of the scale (or supplied external estimate of scale, if option InitialEst is not empty).

p x length(eff) matrix containing MM estimate of regression coefficients for each value of eff.

n x length(eff) matrix containing the monitoring of scaled residuals for each value of eff.

n x length(eff) matrix containing the estimates of the weights for each value of eff

Boolean matrix containing the list of the units declared as outliers for each value of eff using confidence level specified in input scalar conflev.

Confidence level which is used to declare units as outliers. Remark: conflev will be used to draw the horizontal line (confidence band) in the plot.

Number of subsets wihtout full rank. Notice that Notice that singsub > 0.1*(number of subsamples) produces a warning

string identifying the rho function which has been used.

vector which contains the additional parameters for the specified rho function which have been used. For hyperbolic rho function the value of k =sup CVC. For Hampel rho function the parameters a, b and c.

vector containing the value of eff which have been used.

the data matrix X

the response vector y

a single value, interpreted as an integer